Question

What must be added to the following expression to make it a whole square? 

4x² − 12x + 7

Answer 1

Let us consider the following expression: \[4 x^2 - 12x + 7\]

The above expression can be written as: \[4 x^2 - 12x + 7 = \left( 2x \right)^2 - 2 \times 2x \times 3 + 7\]

It is evident that if 2x is considered as the first term and 3 is considered as the second term, 2 is required to be added to the above expression to make it a perfect square. Therefore, 7 must become 9.
Therefore, adding and subtracting 2 in the above expression, we get:

\[\left( 4 x^2 - 12x + 7 \right) + 2 - 2 = \left\{ \left( 2x \right)^2 - 2 \times 2x \times 3 + 7 \right\} + 2 - 2 = \left\{ \left( 2x \right)^2 - 2 \times 2x \times 3 + 9 \right\} - 2 = \left( 2x + 3 \right)^2 - 2\]  Thus, the answer is 2.